Radical of a ring - Revision history

en>Myasuda: /* The Artinian radical */ sp

2014-11-28T21:55:32Z

The Artinian radical: sp

← Older revision		Revision as of 23:55, 28 November 2014
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	~~In [[mathematics]],~~ the ~~[[tensor product]]~~ of ~~two [[algebra (ring theory)\|''R''-algebras]] is~~ also ~~an ''R''-algebra~~. ~~This gives us a '''tensor product~~ of ~~algebras'''~~. ~~The special case ''R'' = '''Z''' gives us a '''tensor product~~ of ~~rings''', since rings may be regarded as '''Z'''-algebras~~.

	~~Let ''R'' be a [[commutative ring]~~] ~~and let ''A'' and ''B'' be [[algebra (ring theory~~)~~\|''R''-algebras]]~~. ~~Since ''A'' and ''B'' may both be regarded as [[module (mathematics)\|''R''-modules]], we may form~~ their ~~[[tensor product of modules\|tensor product]]~~
	:<~~math~~>~~A \otimes_R B~~<~~/math~~>
	~~which is also an~~ ''R''~~-module. We can give~~ the ~~tensor product the structure of an algebra by defining<ref>Kassel (1995)~~, ~~{{Google books quote\|id=S1KE_pToY98C\|page=32\|text=we put an algebra structure on~~ the ~~tensor product\|p~~. ~~32}}~~.</~~ref>~~
	~~:<math>(a_1\otimes b_1)(a_2\otimes b_2)~~ = ~~a_1a_2\otimes b_1b_2</math>~~
	~~and then extending by linearity to all of {{nowrap\|''A''~~&~~thinsp;⊗<sub>''R''</sub>~~&~~thinsp;~~''~~B''}}. This product is easily seen to~~ be ~~''R''~~-~~bilinear, associative, and unital with an identity element given by {{nowrap\|1~~<~~sub~~>~~''A''~~<~~/sub> ⊗ 1<sub~~>'~~'B''</sub>}}~~,~~<ref>Kassel (1995)~~, ~~{{Google books quote\|id=S1KE_pToY98C\|page=32\|text=Its unit is\|p. 32}}~~.<~~/ref~~> ~~where 1~~<~~sub~~>''~~A''~~<~~/sub~~> ~~and 1~~<~~sub~~>'~~'B''</sub> are~~ the ~~identities of ''A'' and ''B''~~. ~~If ''A'' and ''B'' are both commutative then~~ the ~~tensor product is as well~~.

	The ~~tensor product turns~~ the ~~[[category~~ (~~mathematics~~)~~\|category]] of all ''R''-algebras into a [[symmetric monoidal category]]~~.

	There ~~are natural homomorphisms of ''A'' and ''B''~~ to ~~{{nowrap\|''A'' ⊗~~<~~sub~~>~~''R''~~<~~/sub~~>~~ ~~'~~'B''}} given by<ref>Kassel (1995)~~, ~~{{Google books quote\|id=S1KE_pToY98C\|page=32\|text=get algebra morphisms\|p~~. ~~32}}~~.<~~/ref>~~
	~~:<math>a\mapsto a\otimes 1_B</math>~~
	~~:<math~~>~~b\mapsto 1_A\otimes b~~<~~/math~~>
	~~These maps make~~ the ~~tensor product a [[coproduct]] in~~ the ~~category~~ of ~~commutative ''R''-algebras.~~ (~~The tensor product~~ is ~~''not''~~ the ~~coproduct in the category of all ''R''-algebras~~. ~~There~~ the ~~coproduct is given by a more general [[free product]] of algebras).~~

	~~The tensor product of algebras is of constant~~ use ~~in [[algebraic geometry]]: working in the [[opposite category]] to that~~ of ~~commutative ''R''-algebras~~, ~~it provides [[pullback (category theory)\|pullback]]s of [[affine scheme]]s, otherwise known as [[fiber product]]s.~~

	~~<!--Stuf removed from [[ring (mathematics)]]:~~
	~~Since any ring is both a left and right [[module (mathematics)\|module]] over itself, it is possible~~ to ~~construct~~ the ~~[[tensor product of rings\|tensor product]] of ''R'' over a ring ''S'' with another ring ''T''~~ to get ~~another ring provided ''S'' is a central subring of ''R''~~ and ~~''T''~~.
	~~-->~~

	~~==See also==~~
	*[[Extension of scalars]]
	*[[Tensor product of modules]]
	*[[Tensor product of fields]]
	*[[Linearly disjoint]]
	*[[Multilinear subspace learning]]

	~~== Notes ==~~
	~~<references/>~~

	~~== References ==~~
	* {{Citation\| last1=Kassel\| first1=Christian\| date=1995\| title=Quantum groups\| volume = 155\| series=Graduate texts in mathematics \| publisher=Springer\| isbn = 978-0-387-94370-1}}.

	~~{{DEFAULTSORT:Tensor Product Of Algebras}}~~
	~~[[Category:Algebras]]~~
	~~[[Category:Ring theory]]~~
	~~[[Category:Commutative algebra]]~~
	~~[[Category:Multilinear algebra]]~~

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7280488

2013-03-23T05:35:07Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7280488

← Older revision		Revision as of 07:35, 23 March 2013
Line 1:		Line 1:
	~~My name~~ is ~~Michel~~ and ~~I am studying Film Studies~~ and ~~Japanese Studies at Tourinnes~~-La-~~Grosse~~ / ~~Belgium~~.<br><br>~~Feel free~~ to ~~visit my web blog Oltramare~~ ([~~http~~://~~tradercolony~~.~~com click through~~ the ~~up coming web site~~])		{{Refimprove\|date=May 2010}}
			In [[mathematics]], the [[tensor product]] of two [[algebra (ring theory)\|''R''-algebras]] is also an ''R''-algebra. This gives us a '''tensor product of algebras'''. The special case ''R'' = '''Z''' gives us a '''tensor product of rings''', since rings may be regarded as '''Z'''-algebras.

			Let ''R'' be a [[commutative ring]] and let ''A'' and ''B'' be [[algebra (ring theory)\|''R''-algebras]]. Since ''A'' and ''B'' may both be regarded as [[module (mathematics)\|''R''-modules]], we may form their [[tensor product of modules\|tensor product]]
			:<math>A \otimes_R B</math>
			which is also an ''R''-module. We can give the tensor product the structure of an algebra by defining<ref>Kassel (1995), {{Google books quote\|id=S1KE_pToY98C\|page=32\|text=we put an algebra structure on the tensor product\|p. 32}}.</ref>
			:<math>(a_1\otimes b_1)(a_2\otimes b_2) = a_1a_2\otimes b_1b_2</math>
			and then extending by linearity to all of {{nowrap\|''A'' ⊗<sub>''R''</sub> ''B''}}. This product is easily seen to be ''R''-bilinear, associative, and unital with an identity element given by {{nowrap\|1<sub>''A''</sub> ⊗ 1<sub>''B''</sub>}},<ref>Kassel (1995), {{Google books quote\|id=S1KE_pToY98C\|page=32\|text=Its unit is\|p. 32}}.</ref> where 1<sub>''A''</sub> and 1<sub>''B''</sub> are the identities of ''A'' and ''B''. If ''A'' and ''B'' are both commutative then the tensor product is as well.

			The tensor product turns the [[category (mathematics)\|category]] of all ''R''-algebras into a [[symmetric monoidal category]].

			There are natural homomorphisms of ''A'' and ''B'' to {{nowrap\|''A'' ⊗<sub>''R''</sub> ''B''}} given by<ref>Kassel (1995), {{Google books quote\|id=S1KE_pToY98C\|page=32\|text=get algebra morphisms\|p. 32}}.</ref>
			:<math>a\mapsto a\otimes 1_B</math>
			:<math>b\mapsto 1_A\otimes b</math>
			These maps make the tensor product a [[coproduct]] in the category of commutative ''R''-algebras. (The tensor product is ''not'' the coproduct in the category of all ''R''-algebras. There the coproduct is given by a more general [[free product]] of algebras).

			The tensor product of algebras is of constant use in [[algebraic geometry]]: working in the [[opposite category]] to that of commutative ''R''-algebras, it provides [[pullback (category theory)\|pullback]]s of [[affine scheme]]s, otherwise known as [[fiber product]]s.

			<!--Stuf removed from [[ring (mathematics)]]:
			Since any ring is both a left and right [[module (mathematics)\|module]] over itself, it is possible to construct the [[tensor product of rings\|tensor product]] of ''R'' over a ring ''S'' with another ring ''T'' to get another ring provided ''S'' is a central subring of ''R'' and ''T''.
			-->

			==See also==
			*[[Extension of scalars]]
			*[[Tensor product of modules]]
			*[[Tensor product of fields]]
			*[[Linearly disjoint]]
			*[[Multilinear subspace learning]]

			== Notes ==
			<references/>

			== References ==
			* {{Citation\| last1=Kassel\| first1=Christian\| date=1995\| title=Quantum groups\| volume = 155\| series=Graduate texts in mathematics \| publisher=Springer\| isbn = 978-0-387-94370-1}}.

			{{DEFAULTSORT:Tensor Product Of Algebras}}
			[[Category:Algebras]]
			[[Category:Ring theory]]
			[[Category:Commutative algebra]]
			[[Category:Multilinear algebra]]

en>Rschwieb: /* See also */

2012-08-27T14:12:42Z